IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 1 , JANUARY 1998
`
`141
`
`Signal Space Detectors for MTR-Coded Magnetic Recording Channels
`
`Hamid Shafiee, Bernard0 Rub and Robert Kost
`Seagate Technology, 8001 E. Bloomington Freeway, Bloomington, MN 55420
`
`- - 1
`J U --AI--
`
`-
`
`1
`
`
`
`Abstract -In this palper, we present signal space detectors for
`use with maximum transition run (MTR) codes. A three-
`dimensional signal space detector is first derived for an MTR=2
`coded channel. The bit errar rate performance of this detector
`is close to MTR-coded FDTmF(2) throughout the user density
`range of interest. The detector is then modified to be used with
`a time-variant MTR code. Simulation as well as experimental
`results are presented.
`
`I. INTRODUCTION
`
`To improve bit error rate performance, or to increase linear
`recording density, the application of maximum likelihood
`sequence detection (MLSD) to digital magnetic recording
`has been investigated in recent years. It is observed that with
`binary input bits, for MLSD at high recording densities and
`for certain high-ordter partial response channels such as
`E2PRML, the dominant error events are of the form f{+2, -2,
`+2}. A new class of codes called maximum transition run
`(MTR) codes have recently been proposed as a way of
`removing such dominant error events and, hence, increasing
`the minimum Euclidean distance [l]. An MTR-2 code limits
`the maximum number of consecutive transitions to two, and
`consequently, removes all pattems which cause the dominant
`error events (Fig. 1). MTR codes of rate 6/7 have been
`developed in [2] and [3].
`Utilizing the MTR constraint, Brickner and Moon have
`developed an efficient detector called 3D-110 whose
`performance is comparable to fixed delay tree search with
`decision feedback of depth 2 (FDTSIDF(2)) at high symbol
`densities [4]. The detector is constructed by considering
`vectors of received samples in a 3-dimensional space. Using
`three planer boundaries, the signal space is divided into two
`regions each of which correspond to a decision of +1 or -1.
`The 3D-110
`forward
`filter
`removes
`the precursor
`intersymbol interference (ISI) terms and forces the two post-
`cursor IS1 terms to be 1 and 0, respectively, where the cursor
`is also normalized to 1. The feedback filter removes all but
`two post-cursor IS1 terms. With no error propagation, the
`equivalent discrete-time channel pulse response can be
`denoted as "1 10". Such a constraint on the channel response
`is used to further simplify the detector structure.
`While the magnetic channel "natural" response is close to
`the "1 10" target at high recording densities, it deviates from
`the desired target at lower densities. Constraining the pulse
`response to this particular target will then result in
`performance degradation compared to FDTS/DF (2). Even at
`
`(a)
`@)
`Fig. 1: Error events of the form f(2, -2, 2) are caused when (a) a tribit is
`shifted or (b) when a quadbit is mistaken as a dibit or vice versa. MTR-2
`codes remove all pattems containing 2 or more transitions. Timevariant
`MTR codes allow tribits to start at alternate t h e intervals.
`high densities, other factors such as the use of constrained-
`length finite impulse response (FIR) filters may cause
`deviation of the channel response from the "110" target. In
`the next section, we extend the geometrical detection
`approach in [4] and develop a sub-optimal three-dimensional
`signal space detector referred to as 3D-SSD, which does not
`constrain the channel response to any specific target.
`However, it uses the MTR constraint as well as modified
`signal space decision boundaries to simplify the detector
`structure. The new detector provides marginal improvements
`over 3D-110 at higher densities, but provides considerable
`gains at lower densities.
`The dominant error events mentioned above can also be
`removed using a time-variant transition run constraint that
`allows tribits to only start at even- (or odd-) numbered time
`intervals (Fig. 1) [5][6]. Such a relaxed constraint would
`then allow the development of codes with higher rates. Both
`3D-110 and 3D-SSD channels are derived based on the
`assumption that no tribits are allowed in the input sequence.
`In Section 111, we develop modified 3D detectors for use
`with a time-variant MTR code. Bit error rate (BER) results
`from an experimental set-up are included in Section IV.
`
`II. DERIVATION OF THE 3D-SSD CHANNEL
`
`We first consider an MTR-2 coded channel. In Fig. 2, a
`generic block diagram of a digital recording channel which
`uses decision feedback is depicted. Like other decision
`feedback techniques, 3D-SSD uses a whitened matched filter
`to remove all the IS1 terms and to whiten the noise at the
`detector input. Unlike 3D-110, no constraints are enforced
`on the channel coefficients and, hence, the post-cursor IS1
`
`Fig. 2: Schematic dk%" of a decision feedback read channel
`
`Feedback U Nlter
`
`Manuscript received June 16, 1997.
`H. Shafiee, 612-806-256:2, Hamid-R-Shafi@notes.seagate.com; B.
`Rub, 612-806-2933, Bemardo-Rub@notes.seagate.com; R. Kost, 612-806-
`277 1, Robert-E-Kost@nolss.seagate.com.
`
`0018-9464/98$10.00 0 1998 IEEE
`
`UMN EXHIBIT 2018
`LSI Corp. et al. v. Regents of Univ. of Minn.
`IPR2017-01068
`
`
`Page 1 of 6
`
`

`

`142
`
`(1)
`
`"natural" values. The
`terms are allowed to take on the
`feedback filter removes all but two, post-cursor IS1 terms.
`Assuming all the previous decisions are correct, the
`equivalent discrete-time channel response includes three
`terms and is denoted as (l,fi,f2) where, without any loss of
`generality, the main tap is normalized to one. At time k, the
`noiseless input to the decision device, y k , could be written
`as:
`
`( U k - 2 , U k - , 2 ak 1
`I m k X
`(+l, +1, +1)
`0
`1
`(+l, +1, -1)
`(+l, -1, +1)
`2*
`3
`(+l, -1, -1)
`4
`(-1, +1, +1)
`*
`(-1, +1, -1)
`5 .
`Yk = ' A + A u k - 1 + f l a k - ' 2
`(-1, -1, +1)
`6
`where ak is the input data bit at time k.
`(-1. -1. -1)
`7
`I
`The 3D-SSD detector is designed by first considering the
`I
`Table 1: Input dab Pattems and the CofiesPnhg Symbols in the
`symbol constellation in a three-dimensional space. The
`signal space. *Cases 2 and 5 violate the h4TR constraint with
`detector decision depends on the symbol which is closest to
`= -1 and +1, respectively.
`A u ~ - ~
`the vector of observation samples at each time interval. This
`is analogous to finding the path With the minimum Euckkan and k-2 with the IS1 due to the available decisions (i.e., 6k-3
`distance between the observed and desired samples values and Gk-,, at time k) canceled. Notice that at each time k in the
`for fixed-delay detectors such as FDTS/DF or look-ahead
`principle, each pair of detection process, the detector needs to make a decision on
`partial response channels [7,81.
`symbols which point to different detector decisions need to
`the input bit, ak-2
`be separated by a boundary plane. The planar boundaries are
`Table 1 lists input write currents and the corresponding
`combined by a logic rule so that the signal space is noiseless points in the (yk ,Y;-~ ,~f-~)signal space. Notice
`that, depending on the value of ikm3,
`partitioned into two regions, one corresponding to a decision
`either symbol 2 or 5 is
`of + 1 and the other - 1. Depending on where the vector of the disallowed since it represents the present of a tribit, which is
`received symbols falls in the vector space, a binary decision disallowed by
`the MTR=2 code. For example, with
`is released as the detector output. The detector structure is
`A ak-3 = +1, symbol 5 corresponds to a sequence of the form
`simplified by eliminating planes which are redundant or
`separate symbols which are much farther apart than the (+I, -1, +1, -1) which contains three consecutive transitions.
`minimum Euclidean distance.
`Fig. 3(a) and (b) show the symbol constellation for a
`The 3-dimensional observation vector falls in the Lorentzian channel at a symbol density of 2.25 with
`respectively. The
`(yk ,yk-l ,yk-2) signal space. The derivation of the planer &-3 = +1 and &-3 = -1,
`boundaries is simplified with a
`linear vector space extends out of the surface of the paper. Symbols which
`transformation [4]. Lei us define the following parameters corresponds to y:-l = +1 and -1 are denoted by x's and o's,
`Yk =' ak ' f l a k - 1 ' fZak-2
`respectively. The index below the symbol marker points to
`which comprise the axes of the new vector space:
`(2) the corresponding input data patterns listed in Table 1.
`Yi-1 = ak-l + ha,-,
`(3)
`(4)
`Yf-'2 = ak-2
`Here, YL-1 and Yf-2 denote the detector inputs at times k-l
`
`To keep the detector structure simple, we limit the number
`of slicer planes in 3D-SSD to three. The directions of these
`planes are also constrained to further simplify the detector
`
`-2
`
`-1
`
`1
`
`2
`
`0
`0
`yk
`yk
`Fig. 3: (a) Symbol constellation with 6k-3 = -1 , (b) symbol constellation with 6k-3 = +1 . The dark lines show the intersection of the
`boundary planes with the yky;-I surface.
`
`-2
`
`-1
`
`1
`
`2
`
`
`Page 2 of 6
`
`

`

`structure. Initially, four decision planes denoted as A, B, C
`and D, are considered. (Later on, C and D are combined to
`form E.) Let us first consider plane A which separates
`symbols 0 and 4 (as well as 1 and 5 in Fig. 3(a)). Optimal
`decision boundaries are planes bisecting the line which
`connects pairs of symbols of
`interest. However, the
`constrained optimization here, locates a plane which
`separates, not the two symbols, but their projections on the
`surface. ThLe intent is to pick the two coordinates
`Y;...~
`which contribute the most to the distance between the two
`symbols. Clearly, the y[-2 coordinate needs to be retained,
`since the two symbols which correspond to different
`decisions on ak-2 , are well separated on this axis. Of the
`two remaining, except for very low symbol densities
`(Ds<1.6), y;-l contributes more significantly to the distance
`than Y k ‘
`The slicer plane A, therefore, is constrained to only rotate
`perpendicular to the y;-l y[-2 surface. The projection of this
`plane onto the yL-lyt-2 surface will be a line whose
`direction changes as the slicer plane is allowed to rotate. All
`points on the desired line have the same distance from the
`projection of the pair of symbols. Since the coordinates of
`the projections of symbols 0 and 4 on the y;-l yga2 surface
`are given by (1 + f, ,+ 1) and (1 - f, ,- 1) , respectively, the
`equation of the plane A can be obtained by writing:
`cy;-, - (1 + f, N2 + ( Y [ - 2 - = (Yl-1 - (1 - f, HZ + (ye2 +
`This expression could be simplified to yield
`Y i - 2 + A Y k - 1 - f, =
`Using a similar procedure, the equation for slicer B which
`separates symbols 3 and 7 (as well as 2 and 6 in Fig. 3(b)),
`can be found to be:
`Y l - 2 +AY;-, + A =o
`Plane C separates symbols 3 and 5 when $k-3 = -1. Here,
`this plane is constrained to only rotate perpendicular to the
`yk Y[-~
`surface since the two coordinates which contribute
`more significantly to the distance in this case are yk and
`yZ2 . The plane equation can be derived by finding the line
`that bisects the projections of the two symbols on the
`yk Y[-~ surface. The operation is repeated for plane D which
`separates symbols 2 and 4 with ikd3
`= +l. Applying the
`procedure outlined above results in the following four
`boundary equations:
`A Sgn(Yr-2 + fi.v;-1 - fi 1
`sgn(Yr-2 + f,.v;-, + A )
`c: sgn(y[-2 - (J, - $2
`‘k-3 = +’
`)Yk - (fi - f 2
`D: sgn(yr-2 - (1; - f2 )Yt + (fi - fi ))*
`Boundaries C and D could be combined to give:
`E: sgn(y[-’_; - (A - fi )Yk + (fi - f2 )‘k-3)
`The above equation can be further simplified by setting
`(fi - f2)equal to 1, This simplification has a negligible
`
`-1
`
`Gk-3
`
`143
`
`effect on the detector performance since at lower channel
`densities of interest, the two symbols to be separated by this
`plane are farther apart than those separated by planes A and
`B. Therefore, a slight change in the plane orientation and
`position would not impact the relative location of the
`received samples with respect to this plane. The new slicer
`plane becomes:
`E: sgn(yf-2 - yk + ‘k-3
`Substituting for y;-l and yr-2 using Equations (3) and (4),
`the following relations are obtained for the three decision
`planes:
`
`(5)
`(6)
`(7)
`
`A sgn(yk-2 + f,Yk-1 +
`B sgn(yk-2 + AYk-1 +
`E: s d Y k - 2 - Yk +
`where offset values AA, AB and AE are given by
`= (-f, - fifi I‘k-3 - f 2 ‘k-4 - fi
`(8)
`AB = (-fi - fif2>‘k-3 - f 2 ‘k-4 + fi
`(9) .
`AI!? = (-fi + 1)&3 - f2 ‘k-4
`(10)
`The offset levels can, in general, be implemented as short
`FIR filters with binary inputs, 2-input multiplexers or look-
`up tables.
`To arrive at the decision logic, one can move a test point
`through the signal space and record the relative position of
`the point with respect to the planes. The corresponding
`detector output is also noted by finding the closest symbol in
`the constellation to the test point. A logic rule is found by
`combining the cases which result in the same output
`decision. For the three-dimensional case considered here,
`however, the logic rule can be written by inspection (Fig. 3).
`Mapping boundary decisions -1 to 0, the logic rule can be
`written as:
`
`‘k-2 = B.E+ A
`(1 1)
`The 3D-SSD architecture is shown in Fig. 4. With offsets
`implemented using multiplxers, the detector shown in Fig. 4
`uses one multiplier, three slicers, three adders and three two-
`input multiplexers. The 3D-110 detector, on the other hand,
`can be implemented using three slicers, three adders and two
`AE
`I
`
`Fig. 4 3D-SSD detector
`
`
`Page 3 of 6
`
`

`

`144
`
`two-input multiplexers [3].
`In Fig. 5, the simulation results which compare the
`required SNR for a BER of le-4 for a Lorentzian channel
`with additive white Gaussian noise (AWGN) are depicted for
`several detectors. The noisy readback waveform is run
`through a 4th-order Butterworth low-pass front filter. The
`low-pass filter cut-off frequency is set at the Nyquist
`frequency. To remove the influence of filters lengths, both
`forward and feedback filters have a sufficient number of
`taps. The feedback detectors use an MTR=2 code of rate 6/7.
`Results for a (0, 4/4) EL-coded PRML channel are also
`plotted for comparison. The plot shows that the performance
`of 3D-110 approaches that of FDTS/DF(2) as linear density
`increases. At low densities, however, constraining the
`channel response to the "110" target results in performance
`degradation compared
`to FDTS/DF due
`to noise
`enhancement and coloration. The 3D-SSD detector performs
`close to FDTS/DF throughout the user density range. At a
`user density of 2,3D-SSD comes within 0.3 dB of FDTS/DF
`while 3D-110 has about 1.5 dB degradation. As the user
`density is lowered, the dominant error events for MLSD
`starts to change from tribit patterns to single bit events.
`Therefore, the coding gain of the MTR code is reduced.
`
`Required SNR for a BER of l e 4
`
`24 .. . . . . . . . . . .' . . . . . '
`
`2
`
`2 25
`
`2 b
`User Density
`Fig. 5: Required SNR for a BER of le-4 vs. user density
`
`2 75
`
`3
`
`III. 3D DETECTOR WITH TIME-VARIANT MTR CODE
`
`3D-110 and 3D-SSD detectors are both constructed by
`talung advantage of the fact that at each time mterval, only
`one of the two symbols 2 or 5 in Table 1 is present in the
`signal space. This is because MTR-2 codes remove one of
`these two symbols at all times. With the time-variant MTR
`coclc, hOWeVCP, it is possible to have both symbols present in
`the signal constellation at every other time interval. The
`structure of the 3D detectors would have to be modified to
`accommodate the change in the code constraint.
`To design signal space detectors which utilize the new
`code, let us consider the FDTYDF tree of depth two. In Fig.
`6(a) and (b), assuming 2k-3 = +1, the detection tree is shown
`when the root is at an odd or even time interval, respectively.
`Here, without any loss of generality, it is assumed that the
`tribits are only allowed to start at even time intkrvals. Notice
`
`that in Fig. 6(a), as in the previous case, either path 2 or 5 is
`disallowed since it violates the code constraint. For example,
`branch 5 is pruned since it points to tribit pattern {+l, -1, +1,
`-1) which starts at an odd time interval.. Therefore, at odd
`times, the situation is identical to the MTR=2 case. On the
`other hand, when the root is an even time interval as in Fig.
`6(b), both branches 2 and 5 are legal.
`To realize the coding gain for an FDTS/DF(2) detector
`with a time-variant MTR code, one can remove the illegal
`path at odd times and restore it at even times. This would
`prevent shifted tribit errors from occurring since the
`erroneous tribit has to start at a forbidden time interval. But
`the presence of both paths 2 and S increases the chance of
`the erroneous section of the tree to be selected. In fact, as the
`density increases, these errors start to wipe out the code-rate
`benefit of the time-variant MTR code.
`Let us extend the branches 2 and S one step further as
`shown in Fig. 6. This does not affect the situation in Fig, 6(a)
`since the extended branches of path 2 are allowed whereas
`those of 5 are disallowed. However, in Fig. 6(b), the top
`branch for path 2 denoted as 2A, is allowed while the bottom
`branch (2B) is disallowed since it corresponds to a tribit
`which start at an odd position. Similarly, only the bottom
`branch of path 5 (e.g., 5B) is allowed. The two symbols 2A
`and 5B correspond to error events of the form f{2, -2,2,2}.
`Therefore, the distance between
`the
`two should be
`considerably greater than the minimum Euclidean
`ce.
`If the distances from the 4-dimensional observation vector to
`these two symbols are used to select the closer symbol, then
`either path 2 or 5 could be completely pruned from the tree.
`Removal of one of the two symbols (or paths) would then
`make the signal constellation resemble those shown in Fig. 3.
`To come up with a boundary decision for the selection of
`either path 2A or SB, let us first write the sample at time k+l
`as:
`Yk+l = ak+l + hak + fiak-1
`(12)
`In the Signal space ( Y k + l Y Y k ,? k-1 , ? ' k - 2
`symbols 2.4 and
`5B are given by(+l+f,-f,,+l-f,+f,,-l+f,,+l)
`and
`(-1 - fi + f2 ,- 1 + fi - f2 ,1- fi ,- 1) , respectively. The
`new boundary plane P bisects the projection of these points
`surface. Assuming that sample yk+l is
`in the ~ ~ + ~ y ' ' ~ - ,
`available at time k, the equation of the planc can be obtained
`by writing the distance of the point
`y' k-2 ) from the
`tWQ points as given below:
`+ (Y' l k - 2 -v =
`( Y k + l - (1 + f, - f 2
`( ( Y k + l - ('l - fi * 7 2 >>' * (y ' k-2 *lIh
`
`(13)
`The decision due to plane P impacts the symbol
`constellation at even times (i.e., when k-3 is even), since it
`removes either symbol 2 or 5. One way to alleviate the
`computational delay due to this boundary is to come up with
`the decision one time interval early. Toward this end, let us
`write Equation (13) one bit interval early. This gives:
`
`
`Page 4 of 6
`
`

`

`Even
`k-2
`
`Odd,
`k-3
`Root
`
`I
`
`k- 1
`
`I
`
`I
`
`k+l
`
`I
`
`k
`
`Odd
`k-2
`
`I
`
`k- 1
`
`I
`
`I
`
`k
`
`k+l
`
`I
`
`Eveq
`k-3
`Root
`
`Figure 6 FDTS/DF tree: (a) root is an odd time interval; (b) root at an even time interval
`plane C apply when 6k-3 = -1 with the root at an odd time
`Computation of y1+2 at time k, requires the availability of
`the sequence (fik-l,tk-2 ,...,
`interval or when P=-lwith
`the root at an even time
`for feedback cancellation,
`interval. On the other hand, the constellation of Fig. 3(b) and
`where N is the length of the feedback filter. However, c?~-~
`the plane D are applicable when 'k-3 = +1 at odd times or
`and ' k - z decisions ;=e not yet available at time k. (Recall
`when P = +1 at even times. This is summarized below:
`that the output of FI)TS/DF(2) at time k is i?k-2 .) We will,
`C: sgn(y;'-, - yk - I), iik-3 = -1 (odd) or P = -1 (even)
`hence, define a new parameter zk where the IS1 due to the
`feedback taps f3 and f4 are not yet subtracted, i.e.,
`(15)
`Zk+2 == Yk+Z + f3ak-1 + f4 'k-2
`For each symbol, the values of 6k-l and 4 - 2 can be taken
`from its own path. For example, the LHS of Equation (13)
`the distance from symbol 2A,
`for which
`denotes
`6k-1 = -1 and 6k-2 = +1 as seen in Fig. 6(b). On the other
`hand, for symbol 5B (i.e., RHS of Equation (13)), L ? ~ - ~ = +1
`and 6k-2 =-1. The: operation above is similar to local
`feedback cancellation in channels such as reduced state
`sequence estimator (RSSE) [9]. Similarly, local feedback is
`used for the value of 6k-2 when y:-l
`is transformed back to
`yk-l . Substituting for Y k + 2 in Equation (14) and using local
`feedback values foI 6k-l and 6k-2 yield the following
`planer equation:
`(-l+(fi +f3 - m l ( l + f i ) ) Z k + 2
`-Yk-1 +fiL =o
`Since the distance between symbols 2A and 5B is much
`greater
`than
`the
`iminimum Euclidean distance,
`the
`multiplicative factor of z ~ + ~
`can be set at -1 with negligible
`impact on performance. Therefore, slicer plane P is given by
`sgn(-zk+Z - Yk-1 + f 2 'k-3
`)
`As mentioned before, with the decision due to plane P
`available at even tirnes, the symbol constellation is made
`similar to those in Fig, 3 and the modified 3D-SSD detector
`can now be constructed. While the equations for all four
`planes, A, B, c and D remam unchanged, the conditions
`under which planes C and D are applied need to be
`modified. The constellation of Fig. 3(a) and the boundary
`
`Fig. 7: JD-SSD detector for use with the time-variant MTR code
`
`AA
`
`i D:sgn(y;'-, - Yk + l),6k-3 = +I (odd) or P = +I (even)
`
`The two planes can be combined to give plane E' as shown
`below:
`(k - 3) odd
`yk-2 - Yk - AE' + ' k - 3 7
`E : {
`(k - 3) even
`yk-2 - Yk -
`p ,
`where AE' = -&c?~-~ - fZCk-4, which again could be
`implemented as an FIR filter or a two-input multiplexer. The
`fiial decision is again given by Equation (1 1). Fig. 7 shows
`the architecture of the modified 3D-SSD detector for a time-
`variant MTR code.
`In Fig. 8, the performance of 3D-110 and 3D-SSD
`detectors of rates 6/7 and 819 are compared. The code-rate
`gain of the time-variant MTR code is seen by the lower S N R
`
`A
`
`
`Page 5 of 6
`
`

`

`,
`
`Channel
`FDTS/DF(2)
`3D-110
`30-SSD
`
`le-14
`
`3.6e-8
`1.8e-8
`
`Table 2: BER performance; *estimate from margin tests
`
`Reauired SNRfor a BERot 10-4
`
`30-110 (BRMTR)
`-**-3DSSD
`(WMTR)
`+ 3D-SSD (W MTR
`
`V. CONCLUSIONS
`
`2
`
`2 25
`
`2 5
`User Uensity
`
`2 75
`
`4
`3
`
`,,,q - . . L . . . . i - . . . , ~. ~.
`In this paper, design of signal space detectors for MTR-
`15 I;
`coded channels was discussed. Since no constraint was
`enforced on the channel response, the detectors could be
`used over a wide user density range. We developed signal
`space detectors for use with MTR=2 as well as time-variant
`MTR codes. With the MTR=2 code, the new detector
`provides a significant gain over 3D-110 at lower user-
`densities. The performance is further improved with a time-
`variant MTR code of higher rate, especially at higher
`densities.
`
`3D detectors
`Fig. 8: Performance of =-coded
`requirement of the modified 3D-SSD detector. The gain is
`more significant at higher recording densities.
`
`IV. EXPERIMENTAL RESULTS
`
`In this section, we present a comparison of the BER
`performance of 3D-110 and 3D-SSD in an experimental
`is
`system. An MTR=2 modulation code of rate 617
`employed. The encoded data is written on a thin-film disk
`using a thin-film head and read back using a soft-adjacent-
`layer (SAL) magneto-resistive (MR) head. A commercial
`PRML IC is used for generating the write current. The write
`precompensation settings are optimized for the PR4 channel.
`The readback waveform is run through an automatic gain
`control (AGC) and a 7th order equiripple-phase filter. On an
`alternate path, a PRML channel is used to generate a
`synchronized clock signal. This clock is utilized to sample
`the filtered readback waveform using an 8-bit analog-to-
`digital converter (ADC). The sampled signal is equalized
`using a 21-tap digital FIR filter. The equalized sequence is
`then run through appropriate combinations of feedback
`filters and decision devices. The forward and feedback filter
`taps were optimized using the least-mean-square algorithm.
`In Table 2, the BER values are listed. Due to the low error
`count at a user density (Du) of 1.7, the BERs are estimated
`from margin tests, wheire a planer boundary is shifted up or
`down from its nominal position. It is noteworthy to point out
`that the minimum BER at this density for 3D-SSD is
`obtained with the planer boundaries at their nominal
`posibons. In the case of 3D-110, however, due to noise
`coloration, the lowest 13ER is obtained when the boundary
`planes are shifted from their nominal locations. The results
`show that, at the higher line'ar density point, the performance
`of FDTS/DF(2), 3D-110 add 3D-SSD are comparable while
`3D-SSD significantly outderforms 3D-110 at the lower
`recording density.
`I I
`
`1
`
`ACKNOWLEDGMENT
`
`The authors would like to acknowledge many valuable
`discussions with J. Moon and B. Brickner of the University
`of Minnesota. Software development by P. Tsang and B.
`Brickner
`for
`the experimental set-up
`is gratefully
`acknowledged. MTR codes used
`in this work were
`developed by P. Tsang.
`
`REFERENCES
`
`[1] J. Moon and B. Bnckner, "Maximum transition run codes for data
`storage systems", IEEE Trans. Magn., vol. 32, no. 5, pp. 3992-3994,
`Sep. 1996.
`[2] K. Tsang," Method and apparatus for implementing codes with
`maximum transition run lengths", US Patent application.
`[3] Brickner and J. Moon, "Design of a rate 6/7 maximum transition run
`code", preprint, to be published in IEEE Trans. Magn.
`[4] B. Brickner and J. Moon, "A high-dimensional signal space
`implementation ofFDTS/DF", IEEE Trans. Magn., vol. 32, no. 5, pp.
`3941-3943, Sep. 1996.
`[51 E'. Tsang and B Rub, "Maximum transition run length codes with
`location dependent constraints", US patent application.
`[6] W. Bliss, "An $19 rate time-vaq4ng trellis code for high density
`magnetic recording'!, Digests of Intermag 1997.
`[7] A. Patel, R. Rutledge and B. So, 'Ferformance data for a six-sample
`look-ahead 1,7 ML detection channel", IEEE Trans. Magn., vol. 29,
`no. 6, pp. 4012-4014, NOV. 1993,
`R. Yamas& et. al., "A 1,7 code EEPR4 read channel IC with analog
`noise whitened detector", Prw. vfTBBCC, 1997, pp. 316 317,
`9. W. M. Bergmann, S . A. Rajput and F. A. M. van de Laar, "On the
`use of decision feedback for simplifying the Viterbi detector", Phillips
`J. Research, vol. 42, no. 4, pp. 399-428, 1987.
`
`181
`
`[91
`
`
`Page 6 of 6
`
`